### Minimum Constraints in FEA – 5. Submodeling by Kotur Raghavan

**General**

Submodeling is a technique
usually adopted in order to fine-tune the stress results at critical locations.
One conducts local analysis with refined mesh by imposing the boundary
conditions from the full or global model. Deformations from the full (coarse)
model are interpolated with the help of element shape functions. This is the
conventional approach and is based on Saint-Venant’s principle. This can be
called as Cut-Boundary-Displacement (CBD) approach. The approach is well
established and promotes productivity enhancements as the need to refine the
mesh in the entire structure is eliminated. However, there are two limitations.
The first one is that the geometry should be identically similar in the global
and the submodel. If there is any change in stiffness the computed stresses
will be in error. In addition, it is
valid strictly for linear analysis.

The above limitations can be
overcome by resorting to one of the two variants. The first alternative is
based on imposing partly CBD and partly Cut-Boundary-Forces (CBD-CBF). The
second approach will make use of cut-boundary forces only (CBF). This article
is concerned solely with the latter approach.

**The Cut Boundary Force (CBF) Approach**

The steps involved are as
follows.

1. Select
the region of interest.

2. On
all the cut boundaries apply the nodal forces from the results of the global
model. If the meshes are identical this is a straight forward step.

3. If
the meshes are not identical, one need to make use of the resultant forces and
moments in the global model and transfer the same to the nodes in the submodel
using multi pint constraints (MPC). For example, the operation RBE3 in either Nastran or ANSYS needs to be invoked.

4. On
all other boundaries apply surface tractions and nodal loads as in the global
model. In addition, apply all the body forces as in the global model.

5. The
submodel will thus be in self-equilibrium. It can be solved by imposing minimum
constraints appropriately.

The foregoing procedure is not recommended
if the goal is just to get more accurate description of the stress field. It is
recommended if there are any changes in geometry or if nonlinear analysis,
particularly elastic-plastic, analysis is needed.

**Illustrative Example**

A trapezoidal cantilever beam
shown in Fig. 1 is chosen as the example problem.

Fig. 1

The figure contains the global
mode. All the necessary details are described. Coarse mesh with linear elements
is adopted.

Fig. 2

The contours of axial stress for
the whole beam and the middle region are shown in Fig. 2. However, these
results are not of importance. The results of importance are the cut boundary
forces and moments on the boundaries of the region of interest. The resultant
forces acting on the two boundaries are given in the box below. The moments are
about the mid-point of the respective lines.

The forces and moments as above will now be used for analyzing a beam with different geometry in the central region. The target configuration is as shown in Fig. 3.

Fig. 3

As can be seen, a groove is
introduced in the original geometry. The central region is the domain for the
submodel. Details of the submodel are given in Fig. 3.

Fig. 4

Quadratic elements with finer
mesh is used in the submodel. The two master nodes distribute the resultant
loads on the (slave) nodes of the respective boundary edges. Uniform pressure
as in the global model is applied. Displacement constraints are imposed on the
two top corner nodes as indicated.

Fig. 5

Axial stresses are as shown in
Fig. 5. The reaction forces at the constrained degrees of freedom are given in
the box above. The values confirm that the loads and also the constraints are
properly applied.

Fig. 6

For the sake of comparison a full
model with the groove was analysed. Quadratic elements with the same mesh
density as in the submodel was used. The results are given in fig. 6. They can
be seen to be in excellent agreement with the submodel results.

**Discussion**

1. An
important useful application of minimum constraints has been demonstrated. The
traditional Submodeling technique (CBD) which does not permit geometry changes
in the submodel. This limitation has been addressed.

2. Change
in geometry implies change in stiffness. Therefore it is possible to carry out
local elastic-plastic analysis using the results of elastic analysis of the
global model.

3. The
size of the groove in the present study was arbitrary. There is no limitation
as far as geometry changes are concerned. This point of is of significance and
comes handy when one is trying to compare different design alternatives. Local
analyses are adequate and one can avoid expensive repeated global analyses.

4. The
traditional Submodeling technique is essentially a productivity enhancement
tool. The CBF approach, as presented here, will serve to achieve further
productivity enhancements.

5. It
needs to be added here that the technique described in this article is not very
well known. It needs to be popularized. Code developers have a role to play
here.

**Source**

The contents of this article are partial
contents of the paper presented in the ANSYS World Conference, 2008 as shown
below.

This approach filters out all higher modes of traction, i.e., those that are self-balancing on the cut faces. As such the sub-model is likely solving a different problem to that of the global model. Provided, however, the cut faces are sufficiently removed from the region of interest then Saint Venant's principle would appear to work. I studies this some years ago for equilibrium finite elements:

ReplyDeletehttps://www.ramsay-maunder.co.uk/downloads/Sub_Modelling_and_Boundary_Conditions.pdf

My intuitive feeling is that the CBF approach is less sensitive to cut boundary locations as compared to the traditional approach. However, I am not still very sure. I will visit the link. I will also carry out parametric studies. Thanks for your observations.

ReplyDeleteIs there a situation where CBD is not possible and CBF is the only approach? In CBD no constraints are needed except if the sub region chosen involves parent constrained nodes

ReplyDeleteCBD will always work as long as the geometry, and hence the stiffness, is same.

ReplyDelete